Here, we will construct an explicit example of this phenomenon for the convenience of the reader.
Let \(\epsilon \in (0,1)\)  be arbitrary and fix \(N\in \)  with \(N\ge 4\)  so that \((\log 3)/(\log N)\le \epsilon \) .
First, for \(t\in (0,1/N)\) , consider the IFS \(\Phi _t=\lbrace S_0,S_t,S_{1-1/N}\rbrace \)  where \(S_j(x)=\frac{x}{N}+j\)  for \(j\in \lbrace 0,t,1-1/N\rbrace \) .
Observe that if \(t\notin \) , then the IFS has no exact overlaps.
Moreover, the same argument as [1]} shows that the IFS \(\Phi _t\)  satisfies the ESC for all \(t\notin \) .
